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%I #16 Sep 08 2022 08:45:30
%S 1,1,8,9,36,43,120,147,329,406,784,966,1680,2058,3312,4026,6105,7359,
%T 10648,12727,17732,21021,28392,33397,43953,51324,66080,76636,96832,
%U 111588,138720,158916,194769,221901,268584,304437,364420,411103,487256,547239,642873
%N Measures of entanglement in 3-qbits.
%D David Meyer and Nolan Wallach, Invariants for multiple qubits: the case of 3 qubits, Mathematics of quantum computing, Computational Mathematics Series, 77-98, Chapman & Hall/CRC Press, 2002.
%H Colin Barker, <a href="/A129548/b129548.txt">Table of n, a(n) for n = 0..1000</a>
%H Nolan Wallach, <a href="http://dx.doi.org/10.1007/s10440-005-0471-3">The Hilbert series of measures of entanglement for 4 q-bits</a>, Acta Appl. Math. 86(2005), 203-220.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
%F a(n) = [x^(2n)] (1+x^4)*(1+x^4+x^8)/((1-x^2)*(1-x^4)^5*(1-x^6)).
%F a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320. - _Luce ETIENNE_, Oct 15 2015.
%F From _Colin Barker_, Oct 15 2015: (Start)
%F a(n) = (n^6+24*n^5+280*n^4+1920*n^3+7504*n^2+14976*n+11520)/11520 (n even).
%F a(n) = (n^6+24*n^5+235*n^4+1200*n^3+3319*n^2+4536*n+2205)/11520 (n odd).
%F G.f.: -(x^2-x+1)*(x^2+1) / ((x-1)^7*(x+1)^5). (End)
%F a(n) = 2*a(n-1)+4*(n-2)-10*a(n-3)-5*a(n-4)+20*a(n-5)-20*a(n-7)+5*a(n-8)+10*a(n-9)-4*a(n-10)-2*a(n-11)+a(n-12) for n>11. - _Wesley Ivan Hurt_, Oct 15 2015
%p A129548:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320: seq(A129548(n), n=0..50); # _Wesley Ivan Hurt_, Oct 15 2015
%t CoefficientList[Series[(x^2 - x + 1)*(x^2 + 1)/((1 - x)^7*(x + 1)^5), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Oct 15 2015 *)
%o (PARI) Vec(-(x^2-x+1)*(x^2+1)/((x-1)^7*(x+1)^5) + O(x^50)) \\ _Colin Barker_, Oct 15 2015
%o (Magma) [(2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320 : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 15 2015
%Y Cf. A000217, A129549.
%K nonn,easy
%O 0,3
%A _Mike Zabrocki_, Apr 20 2007
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